knitr::opts_chunk$set(fig.width = 8,fig.height = 6)
library(tidyverse)
## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ ggplot2 3.3.6      ✔ purrr   0.3.5 
## ✔ tibble  3.1.8      ✔ dplyr   1.0.10
## ✔ tidyr   1.2.1      ✔ stringr 1.4.1 
## ✔ readr   2.1.3      ✔ forcats 0.5.2 
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
library(rayshader)
library(patchwork)
library(skimr)
library(visdat)
library(ggplot2)
library(GGally)
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
library(corrplot)
## corrplot 0.92 loaded

A first inspection of the dataset with 78 variables

importing the data

D = read.table("/Users/macbook/Documents/Bayesian Statistics/Project/Raw_data/LIPIDI/78 variabili/101_lipidi-PreProcessed-IM-Step1-Step2-Step4-Step5-101.txt")
sum(is.na(D))
## [1] 634087

the numbers of na is substantial

 vis_miss(D,warn_large_data = FALSE)
## Warning: `gather_()` was deprecated in tidyr 1.2.0.
## ℹ Please use `gather()` instead.
## ℹ The deprecated feature was likely used in the visdat package.
##   Please report the issue at <]8;;https://github.com/ropensci/visdat/issueshttps://github.com/ropensci/visdat/issues]8;;>.

the missing data is about 45%

skim(D)
Data summary
Name D
Number of rows 18229
Number of columns 78
_______________________
Column type frequency:
numeric 78
________________________
Group variables None

Variable type: numeric

skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
X401.18155749152 1561 0.91 0.14 0.07 0.02 0.09 0.13 0.18 0.81 ▇▂▁▁▁
X402.175372221284 4352 0.76 0.09 0.04 0.02 0.06 0.08 0.11 0.38 ▇▅▁▁▁
X409.217742652953 4174 0.77 0.10 0.11 0.03 0.06 0.07 0.10 1.34 ▇▁▁▁▁
X419.246407498537 12919 0.29 0.06 0.01 0.02 0.05 0.05 0.06 0.14 ▃▇▂▁▁
X426.123469726227 4124 0.77 0.13 0.07 0.03 0.09 0.12 0.15 1.25 ▇▁▁▁▁
X437.311179462526 9143 0.50 0.13 0.05 0.04 0.10 0.12 0.15 0.52 ▇▃▁▁▁
X442.117407247144 8813 0.52 0.22 0.13 0.03 0.14 0.18 0.26 1.37 ▇▂▁▁▁
X447.212994469759 542 0.97 0.62 0.24 0.01 0.45 0.60 0.77 2.14 ▃▇▂▁▁
X448.206629110332 11103 0.39 0.11 0.03 0.03 0.08 0.10 0.12 0.38 ▇▇▁▁▁
X449.181170853303 11116 0.39 0.11 0.04 0.02 0.08 0.10 0.13 0.49 ▇▅▁▁▁
X465.221110403551 9453 0.48 0.05 0.02 0.02 0.04 0.04 0.05 0.25 ▇▁▁▁▁
X478.277226797819 4972 0.73 0.06 0.02 0.01 0.05 0.06 0.07 0.15 ▁▇▅▁▁
X479.248654523272 12046 0.34 0.04 0.02 0.02 0.03 0.04 0.05 0.32 ▇▁▁▁▁
X497.206369201144 10686 0.41 0.07 0.22 0.02 0.05 0.06 0.07 9.15 ▇▁▁▁▁
X506.28595502602 13991 0.23 0.05 0.01 0.01 0.04 0.05 0.06 0.14 ▂▇▂▁▁
X511.153597066032 11218 0.38 0.52 1.12 0.01 0.04 0.05 0.24 11.68 ▇▁▁▁▁
X513.237647290215 14480 0.21 0.08 0.06 0.01 0.04 0.06 0.09 0.77 ▇▁▁▁▁
X524.279211549971 4709 0.74 0.10 0.03 0.02 0.08 0.09 0.12 0.30 ▂▇▂▁▁
X525.261547813485 12627 0.31 0.05 0.01 0.01 0.04 0.05 0.06 0.12 ▁▇▃▁▁
X538.265072270527 9789 0.46 0.05 0.01 0.02 0.04 0.05 0.05 0.10 ▂▇▅▁▁
X552.2715207686 10676 0.41 0.06 0.01 0.01 0.05 0.06 0.06 0.12 ▁▇▇▁▁
X553.239346145671 2177 0.88 0.19 0.09 0.01 0.13 0.18 0.24 0.69 ▅▇▂▁▁
X558.153387011278 14329 0.21 0.06 0.02 0.02 0.05 0.05 0.07 0.26 ▇▂▁▁▁
X566.28121499168 14220 0.22 0.08 0.02 0.02 0.06 0.08 0.09 0.17 ▁▇▆▁▁
X568.285314369757 9344 0.49 0.16 0.05 0.02 0.12 0.15 0.19 0.42 ▂▇▃▁▁
X576.257761688406 10669 0.41 0.06 0.02 0.02 0.04 0.05 0.06 0.23 ▇▅▁▁▁
X577.277943537651 9942 0.45 0.12 0.05 0.03 0.09 0.11 0.14 0.52 ▇▅▁▁▁
X578.280306512124 10705 0.41 0.07 0.03 0.02 0.06 0.07 0.09 0.29 ▇▅▁▁▁
X592.267287982367 12440 0.32 0.08 0.03 0.02 0.07 0.08 0.10 0.26 ▆▇▁▁▁
X595.24167872285 13615 0.25 0.08 0.02 0.03 0.07 0.08 0.09 0.16 ▁▇▆▁▁
X599.312316651348 1947 0.89 0.15 0.06 0.02 0.11 0.14 0.17 0.92 ▇▁▁▁▁
X600.287990847865 9059 0.50 0.08 0.02 0.02 0.06 0.08 0.09 0.31 ▇▇▁▁▁
X614.280680723305 11990 0.34 0.06 0.02 0.02 0.05 0.06 0.07 0.22 ▇▆▁▁▁
X616.374126264944 13264 0.27 0.05 0.01 0.02 0.04 0.05 0.05 0.09 ▁▇▇▂▁
X652.361088301576 9561 0.48 0.08 0.03 0.02 0.06 0.08 0.10 0.21 ▃▇▅▁▁
X653.324143837042 2179 0.88 0.24 0.10 0.01 0.18 0.23 0.29 1.28 ▇▃▁▁▁
X654.321418202011 10592 0.42 0.14 0.04 0.02 0.11 0.14 0.17 0.35 ▁▇▅▁▁
X666.348091732679 10727 0.41 0.12 0.04 0.02 0.09 0.11 0.15 0.31 ▂▇▅▁▁
X668.350119384666 8434 0.54 0.09 0.04 0.02 0.06 0.08 0.11 0.26 ▆▇▃▁▁
X697.336385172531 765 0.96 0.28 0.10 0.02 0.22 0.28 0.35 0.63 ▂▆▇▂▁
X698.342468583488 6941 0.62 0.18 0.06 0.01 0.14 0.18 0.22 0.45 ▂▇▇▁▁
X713.313334386064 7293 0.60 0.21 0.09 0.02 0.16 0.21 0.25 1.06 ▇▅▁▁▁
X723.369767544851 9485 0.48 0.08 0.02 0.01 0.07 0.08 0.10 0.16 ▁▅▇▃▁
X724.357765223416 11747 0.36 0.06 0.02 0.01 0.04 0.06 0.07 0.17 ▂▇▂▁▁
X726.353588041765 10899 0.40 0.15 0.04 0.02 0.13 0.15 0.17 0.48 ▁▇▁▁▁
X737.153366400938 4088 0.78 0.14 0.07 0.01 0.09 0.13 0.17 0.73 ▇▃▁▁▁
X753.367749586126 8284 0.55 0.13 0.04 0.02 0.10 0.13 0.15 0.32 ▂▇▆▁▁
X762.346785523875 12992 0.29 0.08 0.02 0.02 0.06 0.08 0.09 0.20 ▂▇▃▁▁
X764.372631793774 11446 0.37 0.05 0.02 0.01 0.04 0.05 0.07 0.14 ▃▇▃▁▁
X774.982938560717 8359 0.54 0.19 0.06 0.02 0.15 0.19 0.23 0.53 ▁▇▃▁▁
X775.976648338383 13999 0.23 0.08 0.02 0.02 0.06 0.08 0.09 0.17 ▂▇▇▂▁
X778.403735371109 9983 0.45 1.18 0.53 0.02 0.82 1.14 1.52 4.02 ▃▇▂▁▁
X779.39508853726 5855 0.68 0.50 0.26 0.01 0.29 0.46 0.68 1.83 ▇▇▃▁▁
X780.399514391103 8392 0.54 0.27 0.13 0.02 0.17 0.24 0.35 0.97 ▇▇▃▁▁
X793.327494359355 14448 0.21 0.08 0.03 0.01 0.06 0.08 0.10 0.17 ▂▆▇▅▁
X806.438265275953 1282 0.93 1.50 1.55 0.02 0.37 0.99 2.04 10.18 ▇▂▁▁▁
X807.430315603065 2270 0.88 0.93 0.81 0.01 0.27 0.70 1.37 4.61 ▇▃▂▁▁
X808.418052996172 4205 0.77 0.48 0.34 0.01 0.21 0.40 0.70 2.01 ▇▅▂▁▁
X822.416323513071 2717 0.85 1.34 0.84 0.02 0.64 1.25 1.92 5.52 ▇▇▃▁▁
X823.418850404014 3984 0.78 0.87 0.51 0.02 0.48 0.79 1.23 2.88 ▇▇▅▁▁
X836.412896788889 9507 0.48 0.22 0.09 0.01 0.15 0.20 0.28 0.62 ▂▇▃▁▁
X862.437600249503 280 0.98 2.79 2.42 0.01 0.66 2.24 4.31 13.04 ▇▃▂▁▁
X863.420550439 1936 0.89 1.53 1.15 0.01 0.54 1.31 2.31 6.36 ▇▅▃▁▁
X888.444482031706 1435 0.92 9.36 6.70 0.02 3.84 8.39 13.40 40.59 ▇▆▂▁▁
X889.43378336125 4013 0.78 5.00 3.82 0.03 2.09 4.21 7.08 22.78 ▇▅▂▁▁
X890.455559065043 1940 0.89 11.17 8.12 0.03 4.38 9.83 16.57 44.94 ▇▆▃▁▁
X904.460038847805 10652 0.42 2.05 1.79 0.03 0.37 1.77 3.24 9.68 ▇▅▂▁▁
X906.471649609416 673 0.96 12.42 9.56 0.02 4.10 11.08 18.50 51.34 ▇▆▃▁▁
X907.47494585213 8186 0.55 8.65 5.02 0.02 4.82 8.24 12.22 26.62 ▆▇▆▂▁
X908.485206763087 3471 0.81 3.38 2.22 0.01 1.68 3.06 4.85 12.40 ▇▇▃▁▁
X933.449659746903 11508 0.37 0.10 0.03 0.01 0.07 0.10 0.12 0.24 ▂▇▆▁▁
X934.487399616051 13583 0.25 0.28 0.12 0.01 0.20 0.27 0.36 0.77 ▂▇▅▁▁
X942.439351429176 13376 0.27 0.08 0.03 0.01 0.05 0.08 0.10 0.24 ▅▇▃▁▁
X943.432502731421 7473 0.59 0.10 0.04 0.01 0.07 0.09 0.12 0.32 ▃▇▂▁▁
X944.447551703558 10962 0.40 0.07 0.03 0.01 0.05 0.06 0.08 0.29 ▇▅▁▁▁
X965.462816974066 2475 0.86 0.10 0.04 0.01 0.07 0.09 0.12 0.33 ▅▇▂▁▁
X966.464009229031 6153 0.66 0.06 0.02 0.01 0.04 0.05 0.07 0.17 ▅▇▂▁▁
X967.456646158017 9342 0.49 0.04 0.01 0.01 0.03 0.04 0.05 0.10 ▂▇▃▁▁

we observe that the missing data is not uniform in the mz, there are some values for which only 20 - 30% of the pixel have a value, and this tends to be small

we replace the missing data with 0 since it means the data for that mz was under threshold

D0 = D
D0[is.na(D0)] = 0

preliminary plotts

look at the distribution and correlation of the data, beginning with similar mz probably ggpairs is not the most efficient, did not change the labels

ggpairs(D,columns = 1:7)

ggpairs(D,columns = 8:14)

ggpairs(D,columns = 15:21)

ggpairs(D,columns = 16:28)

ggpairs(D,columns = 28:35)

ggpairs(D,columns = 36:42)

ggpairs(D,columns = 43:49)

ggpairs(D,columns = 50:56)

we see a lot of correlation in the data especially below

ggpairs(D,columns = 57:63)

ggpairs(D,columns = 64:70)

ggpairs(D,columns = 70:78)

correlation matrix

cm <- cor(D0)
corrplot(cm, method = "color")

the blue blob is the observation from above

pixels = read.table("/Users/macbook/Documents/Bayesian Statistics/Project/Raw_data/LIPIDI/78 variabili/101_lipidi-PreProcessed-XYCoordinates-Step1-Step2-Step4-Step5-101.txt")
colnames(D0) = substr(colnames(D0),1,4)
colnames(pixels) = c("x","y")

Create the datasets we will need:

Data_long            = as_tibble(data.frame( pixels, D0 ))
max_number_of_pixels = apply(Data_long[,1:2],2,max)

Data_array = matrix(NA,max_number_of_pixels[1],max_number_of_pixels[2])

Data_array = array(NA,c(max_number_of_pixels[1],max_number_of_pixels[2],ncol(D0)))

sum(is.na(D0))
## [1] 0
# there must be a better way to do this, but it's sunday morning, please be patient...
for(k in 1:ncol(D0)){
  for(i in 1:nrow(Data_long)){
  Data_array[Data_long$x[i],Data_long$y[i],k] = D0[i,k]
  }
}

dim(Data_array)
##   x   y     
## 157 178  78
Data_very_long = reshape2::melt(Data_long,c("x","y")) %>% mutate(pixel_ind = paste0(x,"_",y), value_ind = rep(1:nrow(Data_long),ncol(D0)))
Data_very_long = Data_very_long %>% group_by(pixel_ind) %>% mutate(n = row_number()) %>% ungroup() %>% mutate(mz = as.numeric(substr(variable,2,4)))
Data_very_long = reshape2::melt(Data_long,c("x","y")) %>% mutate(pixel_ind = paste0(x,"_",y), value_ind = rep(1:nrow(Data_long),ncol(D0)))

Data_very_long = Data_very_long %>% group_by(pixel_ind) %>% mutate(n = row_number()) %>% ungroup() %>% mutate(mz = as.numeric(substr(variable,2,4)))


# subsampling to get a faster plot and not drain memory
sub_ind = sample(unique(Data_very_long$pixel_ind),1000)
# just to get the gist:
ggplot(Data_very_long %>% filter(pixel_ind %in% sub_ind))+
  geom_path(aes(x = mz, y = value, 
                col=pixel_ind, 
                group = pixel_ind),alpha=.5)+theme_bw()+theme(legend.position = "none")+xlab("m.z")+scale_color_viridis_d(option = "A")+
  scale_x_continuous(n.breaks = 20)

investigating the different peaks

the first mz don’t contain a lot of info, quite noisy

we investigate the spike arround 500

the spike is in mz 511 on the edge
- possible problem of the instrument on the edge of the brain - outlier ? the rest is just noise

spike arround 775

the spike is relative to 778 779 780

spike arround 800 and 825

very similar to each other

after 850

this are all the same, it is the high correlated blob in the correlation matrix

there is another peak around 900

the peak is in 906 907 908 which are very similar

the rest is not that itresting, very low values resamble kind on veramble the previuus structure but basically noise

A comprehensive look at the peaks

exept for the 511 which is in the edge, the peaks are all correlarend and therefore are all the same information

PCA

pca = princomp(D0)
plot(pca)

summary(pca)
## Importance of components:
##                           Comp.1     Comp.2    Comp.3      Comp.4     Comp.5
## Standard deviation     16.063465 3.74669709 1.8050014 1.557047404 1.16570821
## Proportion of Variance  0.917156 0.04989556 0.0115803 0.008617244 0.00482997
## Cumulative Proportion   0.917156 0.96705159 0.9786319 0.987249138 0.99207911
##                             Comp.6      Comp.7       Comp.8       Comp.9
## Standard deviation     0.775473863 0.732621491 0.5247242544 0.4176023644
## Proportion of Variance 0.002137465 0.001907761 0.0009786491 0.0006198558
## Cumulative Proportion  0.994216573 0.996124335 0.9971029837 0.9977228394
##                             Comp.10      Comp.11      Comp.12      Comp.13
## Standard deviation     0.3441918928 0.3285733456 0.2614626699 0.2329424828
## Proportion of Variance 0.0004210814 0.0003837332 0.0002429876 0.0001928689
## Cumulative Proportion  0.9981439208 0.9985276540 0.9987706416 0.9989635105
##                             Comp.14      Comp.15      Comp.16      Comp.17
## Standard deviation     0.1901943823 0.1719970735 1.542005e-01 1.494558e-01
## Proportion of Variance 0.0001285761 0.0001051494 8.451544e-05 7.939449e-05
## Cumulative Proportion  0.9990920865 0.9991972360 9.992818e-01 9.993611e-01
##                             Comp.18      Comp.19      Comp.20      Comp.21
## Standard deviation     1.437853e-01 1.277647e-01 1.156808e-01 1.124762e-01
## Proportion of Variance 7.348417e-05 5.802117e-05 4.756501e-05 4.496621e-05
## Cumulative Proportion  9.994346e-01 9.994927e-01 9.995402e-01 9.995852e-01
##                             Comp.22      Comp.23      Comp.24      Comp.25
## Standard deviation     1.112338e-01 1.101552e-01 9.828352e-02 8.492948e-02
## Proportion of Variance 4.397825e-05 4.312951e-05 3.433414e-05 2.563786e-05
## Cumulative Proportion  9.996292e-01 9.996723e-01 9.997066e-01 9.997323e-01
##                             Comp.26      Comp.27      Comp.28      Comp.29
## Standard deviation     8.148553e-02 7.341267e-02 7.132058e-02 0.0693419808
## Proportion of Variance 2.360075e-05 1.915609e-05 1.807984e-05 0.0000170906
## Cumulative Proportion  9.997559e-01 9.997750e-01 9.997931e-01 0.9998101895
##                             Comp.30      Comp.31      Comp.32      Comp.33
## Standard deviation     6.664939e-02 6.041977e-02 0.0591444918 5.537155e-02
## Proportion of Variance 1.578909e-05 1.297546e-05 0.0000124335 1.089778e-05
## Cumulative Proportion  9.998260e-01 9.998390e-01 0.9998513875 9.998623e-01
##                             Comp.34      Comp.35      Comp.36      Comp.37
## Standard deviation     5.324759e-02 5.195975e-02 0.0500307473 4.844369e-02
## Proportion of Variance 1.007778e-05 9.596191e-06 0.0000088969 8.341404e-06
## Cumulative Proportion  9.998724e-01 9.998820e-01 0.9998908562 9.998992e-01
##                             Comp.38      Comp.39      Comp.40      Comp.41
## Standard deviation     4.698990e-02 4.254707e-02 3.900727e-02 3.613078e-02
## Proportion of Variance 7.848266e-06 6.434341e-06 5.408237e-06 4.640016e-06
## Cumulative Proportion  9.999070e-01 9.999135e-01 9.999189e-01 9.999235e-01
##                             Comp.42      Comp.43      Comp.44      Comp.45
## Standard deviation     3.413272e-02 3.298352e-02 3.136910e-02 3.051343e-02
## Proportion of Variance 4.141012e-06 3.866862e-06 3.497589e-06 3.309381e-06
## Cumulative Proportion  9.999277e-01 9.999315e-01 9.999350e-01 9.999383e-01
##                             Comp.46      Comp.47      Comp.48      Comp.49
## Standard deviation     2.993919e-02 2.931011e-02 2.866060e-02 2.810079e-02
## Proportion of Variance 3.185993e-06 3.053513e-06 2.919680e-06 2.806738e-06
## Cumulative Proportion  9.999415e-01 9.999446e-01 9.999475e-01 9.999503e-01
##                             Comp.50      Comp.51      Comp.52      Comp.53
## Standard deviation     2.706671e-02 2.659691e-02 2.635536e-02 2.555264e-02
## Proportion of Variance 2.603969e-06 2.514359e-06 2.468896e-06 2.320792e-06
## Cumulative Proportion  9.999529e-01 9.999554e-01 9.999579e-01 9.999602e-01
##                             Comp.54      Comp.55      Comp.56      Comp.57
## Standard deviation     2.524454e-02 2.487671e-02 2.455916e-02 2.438560e-02
## Proportion of Variance 2.265165e-06 2.199636e-06 2.143836e-06 2.113642e-06
## Cumulative Proportion  9.999625e-01 9.999647e-01 9.999668e-01 9.999689e-01
##                             Comp.58      Comp.59      Comp.60      Comp.61
## Standard deviation     2.377389e-02 2.311280e-02 2.300476e-02 2.231907e-02
## Proportion of Variance 2.008932e-06 1.898759e-06 1.881049e-06 1.770585e-06
## Cumulative Proportion  9.999709e-01 9.999728e-01 9.999747e-01 9.999765e-01
##                             Comp.62      Comp.63      Comp.64      Comp.65
## Standard deviation     2.215556e-02 2.197574e-02 2.183727e-02 2.111930e-02
## Proportion of Variance 1.744738e-06 1.716532e-06 1.694967e-06 1.585346e-06
## Cumulative Proportion  9.999782e-01 9.999800e-01 9.999817e-01 9.999832e-01
##                             Comp.66      Comp.67      Comp.68      Comp.69
## Standard deviation     2.065927e-02 2.060456e-02 2.037009e-02 1.987790e-02
## Proportion of Variance 1.517033e-06 1.509007e-06 1.474860e-06 1.404448e-06
## Cumulative Proportion  9.999848e-01 9.999863e-01 9.999877e-01 9.999891e-01
##                             Comp.70      Comp.71      Comp.72      Comp.73
## Standard deviation     1.939794e-02 1.925609e-02 1.916565e-02 1.876678e-02
## Proportion of Variance 1.337445e-06 1.317955e-06 1.305605e-06 1.251827e-06
## Cumulative Proportion  9.999905e-01 9.999918e-01 9.999931e-01 9.999944e-01
##                             Comp.74      Comp.75      Comp.76      Comp.77
## Standard deviation     1.863689e-02 1.821375e-02 1.753698e-02 1.738131e-02
## Proportion of Variance 1.234558e-06 1.179135e-06 1.093136e-06 1.073816e-06
## Cumulative Proportion  9.999956e-01 9.999968e-01 9.999979e-01 9.999989e-01
##                             Comp.78
## Standard deviation     1.727547e-02
## Proportion of Variance 1.060778e-06
## Cumulative Proportion  1.000000e+00

the pca works well because of the correlation

PCA1 = ggplot(Data_long %>% mutate(pca1 = pca$scores[,1]))+ theme_bw()+
  geom_tile(aes(x=x,y=y,fill = pca1))+scale_fill_viridis_c(option = "A",na.value = "red")+
  theme_void()+theme(legend.position = "bottom")
PCA2 = ggplot(Data_long %>% mutate(pca2 = pca$scores[,2]))+ theme_bw()+
  geom_tile(aes(x=x,y=y,fill = pca2))+scale_fill_viridis_c(option = "A",na.value = "red")+
  theme_void()+theme(legend.position = "bottom")

PCA1+PCA2

Note: we are always using the same info

if we invert the colormap

PCA1v2 = ggplot(Data_long %>% mutate(pca1 = pca$scores[,1]))+ theme_bw()+
  geom_tile(aes(x=x,y=y,fill = -1*pca1))+scale_fill_viridis_c(option = "A",na.value = "red")+
  theme_void()+theme(legend.position = "bottom")

PCA1v2 + P6

the data is clearly dominated by this peaks that are highly correlated and spatially correlated as well

Plotting the principal components

first component

second component

do we need to be smarter and explore the non dominant factors or is enough?

sessionInfo()
## R version 4.2.1 (2022-06-23)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Big Sur ... 10.16
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] it_IT.UTF-8/it_IT.UTF-8/it_IT.UTF-8/C/it_IT.UTF-8/it_IT.UTF-8
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] corrplot_0.92     GGally_2.1.2      visdat_0.5.3      skimr_2.1.4      
##  [5] patchwork_1.1.2   rayshader_0.24.10 forcats_0.5.2     stringr_1.4.1    
##  [9] dplyr_1.0.10      purrr_0.3.5       readr_2.1.3       tidyr_1.2.1      
## [13] tibble_3.1.8      ggplot2_3.3.6     tidyverse_1.3.2  
## 
## loaded via a namespace (and not attached):
##  [1] httr_1.4.4          sass_0.4.2          viridisLite_0.4.1  
##  [4] jsonlite_1.8.2      foreach_1.5.2       modelr_0.1.9       
##  [7] bslib_0.4.0         assertthat_0.2.1    highr_0.9          
## [10] googlesheets4_1.0.1 cellranger_1.1.0    yaml_2.3.6         
## [13] progress_1.2.2      pillar_1.8.1        backports_1.4.1    
## [16] glue_1.6.2          digest_0.6.30       RColorBrewer_1.1-3 
## [19] rvest_1.0.3         colorspace_2.0-3    plyr_1.8.7         
## [22] htmltools_0.5.3     pkgconfig_2.0.3     broom_1.0.1        
## [25] haven_2.5.1         scales_1.2.1        tzdb_0.3.0         
## [28] googledrive_2.0.0   farver_2.1.1        generics_0.1.3     
## [31] ellipsis_0.3.2      cachem_1.0.6        withr_2.5.0        
## [34] repr_1.1.4          cli_3.4.1           magrittr_2.0.3     
## [37] crayon_1.5.2        readxl_1.4.1        evaluate_0.17      
## [40] fs_1.5.2            fansi_1.0.3         doParallel_1.0.17  
## [43] xml2_1.3.3          tools_4.2.1         prettyunits_1.1.1  
## [46] hms_1.1.2           gargle_1.2.1        lifecycle_1.0.3    
## [49] munsell_0.5.0       reprex_2.0.2        compiler_4.2.1     
## [52] jquerylib_0.1.4     rlang_1.0.6         grid_4.2.1         
## [55] iterators_1.0.14    rstudioapi_0.14     htmlwidgets_1.5.4  
## [58] labeling_0.4.2      base64enc_0.1-3     rmarkdown_2.17     
## [61] gtable_0.3.1        codetools_0.2-18    reshape_0.8.9      
## [64] DBI_1.1.3           reshape2_1.4.4      R6_2.5.1           
## [67] lubridate_1.8.0     knitr_1.40          fastmap_1.1.0      
## [70] utf8_1.2.2          stringi_1.7.8       parallel_4.2.1     
## [73] Rcpp_1.0.9          vctrs_0.4.2         rgl_0.110.2        
## [76] dbplyr_2.2.1        tidyselect_1.2.0    xfun_0.33